Matrices play a fundamental role in the so-called modem theory of
linear control systems, and interest in this field has been growing
considerably in recent years. In consequence a large number of new results
has been published in journals and reports, and this often means that
valuable time has to be spent searching through the literature for some
required fact. The primary aim of this book is to try to reduce this
rather wasteful expenditure of effort by presenting a collection of some
of the more interesting and useful recent developments in the
applications of matrices to control theory and related topics, especially linear
programming.
Author(s): Barnett, S.
Publisher: Van Nostrand Reinhold Co.
Year: 1971
Language: English
Pages: 234
Tags: Математика;Методы оптимизации;
Barnett,S.Matrices in Control Theory with applications to linear programming (VNR,1971) ......Page 3
Copyright ......Page 4
Contents ......Page 11
Preface vii ......Page 6
List of Symbols xi ......Page 10
1.1 Definitions and basic properties. 1 ......Page 13
1.2 Matrices having relatively prime determinants. 7 ......Page 19
1.3 Degrees of greatest common divisors of invariant factors of two regular matrices. 12 ......Page 24
1.4 Linear system theory; relatively prime matrices. 15 ......Page 27
1.5 A simple class of parametric linear programming problems. 22 ......Page 34
2.1 Resultant and discriminant. 29 ......Page 41
2.2 Matrix expressions. 35 ......Page 47
2.3 Location of zeros of a complex polynomial. 44 ......Page 56
3.1 McMillan form of a rational matrix. 56 ......Page 68
3.2 Realization theory. 58 ......Page 70
3.3 Rosenbrock’s theory of linear systems. 64 ......Page 76
3.4 Positive-real matrices and spectral factorization. 72 ......Page 84
4.1 Real stability matrices and Liapunov theory. 84 ......Page 96
4.2 Inertia of complex matrices. 94 ......Page 106
4.3 Qualitative stability. 98 ......Page 110
5.1 General matrix Riccati differential equations. 108 ......Page 120
5.2 Riccati differential equation in optimal control. 113 ......Page 125
5.3 Riccati algebraic equations. 119 ......Page 131
6.1 The Moore-Penrose inverse. 130 ......Page 142
6.2 Other definitions and properties. 136 ......Page 148
6.3 Application to linear programming. 143 ......Page 155
7.1 Solution of linear equations in integers. 148 ......Page 160
7.2 Transportation problems. 153 ......Page 165
7.3 Combinatorial equivalence. 159 ......Page 171
1 Notes on algebra. 165 ......Page 177
2 Controllability and observability of linear systems. 169 ......Page 181
3 Stability definitions and theorems. 171 ......Page 183
4 Optimal control theory. 173 ......Page 185
5 Additional bibliography. 175 ......Page 187
Solutions to Exercises 178 ......Page 190
Subject Index 215 ......Page 227
Author Index 219 ......Page 231
cover......Page 1
